Urban Transport XVIII
685
Effect analysis on heavy tuned mass damper
system used in urban transportation
L. Li1, C. Geng1 & Q. Yu2
1
Railway and Urban Mass Transit Research Institute,
Tongji University, China
2
Shanghai Tunnel Engineering and
Rail Transit Design and Research Institute, China
Abstract
A tuned mass damper (TMD) is an effective structure. When the mass of TMD is
bigger, the effect to vibration reduction is better. In order to study the heavy
tuned mass damper (HTMD) vibration reduction effect, three methods have been
covered in this paper. The first is to analyze its natural frequencies and mode
shapes. By comparing HTMD with common track, the performance of moderate
and high frequency vibration attenuation has been verified. The second is the
finite element study of wheel load drop on the track structure to simulate the real
wheel impact. The stability and performance of damping property under impact
load has been confirmed. Meanwhile, through dynamic load, parameter
responses of tracks have been obtained. Comparing the track which uses HTMD
with the common one, it can be concluded that the HTMD system fixed on the
rail is effective.
Keywords: heavy tuned mass damper, natural frequency, wheel load drop,
dynamic response.
1 Introduction
Rail transit in China has been developing very rapidly in the past dozens of
years. Because most urban rail transits pass through downtown, the vibration and
noise has a great impact on the environment. The vibration and noise directly
affects people’s living conditions and health, especially to those who live and
work along the lines. A lot of research work has been undertaken on rail transit
WIT Transactions on The Built Environment, Vol 128, © 2012 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
doi:10.2495/UT120581
686 Urban Transport XVIII
vibration noise control worldwide. And successful experience and useful results
have been gained.
Vibration and noise control is one of the hot spots of the present research [1–
4]. According to whether energy input is needed from the outside world or not, it
is divided into two methods, these are active control and passive control.
Theoretically, active control is better than passive control. However, owing to
the theory and engineering practice conditions, most mature technology is
passive control at present in engineering applications. Tuned mass dampers are
one of the most typical applications. Tuned mass dampers have the role of
stabilization to the fierce spot caused by harmonic vibration. It can use
components to suppress the vibration. Even in the most difficult conditions it
also can have the role of vibration reduction.
2 Tuned mass dampers and vibration principle
A TMD is a vibration system composed of a spring, damper and mass, also
known as an active mass damper or harmonic shock absorber, which is a second
vibration system installed in the main vibration system. Through the proper
regulation subsystem of the vibration characteristics, the vibration energy of the
primary system can transfer to the subsystem, and consume in one of the
damping links [5].
Figure 1:
Two-degree freedom model of TMD.
Figure 1 is a common two-degree freedom vibration model for TMD. Among
them, as the object of vibration control the main system is composed of a mass
M and spring K . The natural frequency is 1 . In order to simplify the model,
main system damping is not considered. TMD is built by a mass m, spring k
and damping c being an additional system whose natural frequency is n . The
system of the motion equation is
Mx1  c( x1  x 2 )  Kx1  k ( x1  x2 )  F
mx2  c( x 2  x1 )  k ( x1  x2 )  0
WIT Transactions on The Built Environment, Vol 128, © 2012 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
(1)
Urban Transport XVIII
687
Considering the simple harmonic load F ( t )  F0 e jt , the response can be
expressed as
(2)
x1  X 1e jt ， x 2  X 2 e jt
The amplitude of main vibration system is deduced as
 k  m   c 
2 2
X1 

   mk 2 
2
2
2
  K   M  m   2  c 
2
F0
(3)
Introducing the following items:
F
c
m
Mass ratio  
; damping ratio  
; static deformation X st  0 ;
K
M
2 mk
forced vibration frequency ratio  

n
Then the result will be as follows:
X1

X st
; natural frequency ratio  
n
      2 
      1  1       2 
2 2
2
1   2  2   2

n
2
2
2
2
2 2
2
(4)
Eqn (4) is the vibration rate for amplitude of the main system using TMD. If
the mass ratio and natural frequency are given, amplitude can be calculated. If
we make  and  be a certain value separately, according to the different
damping ratio, the vibration ratio of amplitude curves can be plotted. If damping
is infinite, it is equivalent that TMD is fixed on the main vibration system. It
becomes a single freedom without damping, and resonant amplitude is infinite;
when damping is 0, the TMD makes the resonance frequency divide into two
new resonance frequencies, whose amplitude is still infinite. Then between 0 and
infinite, there muse be a best damping value. There are two crossing points such
as P and Q in the amplitude curves. The position of these two points is not
affected by damping. The amplitude of the main system should be the minimum.
If we make the two points be at the same height, and let them be the peak of the
curves, the TMD will be the best system that we can hope for.
When   0 and    , the results can be
X1
X st

 0
X1
X st
 2  2
1   2  2   2    2 2
(5)
1
1  1     2
(6)

 
WIT Transactions on The Built Environment, Vol 128, © 2012 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
688 Urban Transport XVIII
The points P and Q are located at the cross point of the vibration ratio of the
amplitude curves when   0 and    . Their positions are not affected by
damping, while they are affected by the ratio of natural frequency. When the
mass of TMD is bigger, the effect to vibration reduction is better.
Therefore, in this paper the object is to focus on the HTMD system which is
fixed on the rail (Fig. 2).
Figure 2:
Heavy tuned mass damper.
3 Modal analysis
According to the above theoretical analysis, we can analyze the natural
frequencies of the common track and the track with HTMD on the rail. The main
parameters are shown in Table 1.
Table 1:
Common track and track with TMD on the rail.
Reference value
60
2
77.45
cross-section area ( cm )
210
elastic modulus ( GPa )
passion ratio
0.3
25
calculated length ( m )
6, 25, 50
fastener
stiffness ( kN / mm )
damping ( N  s / m )
7.5e 4
HTMD
length between two rails: 1.35m;
length outside of rail: 0.5m;
thickness: 120mm
spring between HTMD and rail ( kN / mm )
200e 6
rail
Parameters
size ( kg / m )
WIT Transactions on The Built Environment, Vol 128, © 2012 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
Urban Transport XVIII
Table 2:
689
Natural frequencies of common track and the one with HTMD.
Nodal
stiffness
( kN / mm )
6
common
25
50
6
HTMD
25
50
1st
( Hz )
2nd
( Hz )
3rd
( Hz )
4th
( Hz )
5th
( Hz )
6th
( Hz )
65.028
132.5
187.02
18.821
25.609
35.828
65.335
132.67
187.15
22.071
28.028
37.504
66.372
133.25
187.58
26.657
31.681
40.165
68.62
134.61
188.6
32.049
36.231
43.656
72.398
137.19
190.55
33.069
38.116
47.009
78.214
141.39
193.83
37.943
40.228
47.821
From the data in Table 3, it can be seen that the natural frequencies of the rail
with HTMD is smaller than those of common track, which explains that tuned
mass has a certain effect on vibration control.
Table 3:
Vertical displacement of rail (mm).
common track
HTMD
fastener stiffness（kN / mm）
6
25
50
1.245
1.06
0.899
0.644
0.623
0.599
The natural frequencies of the track structure vary with the stiffness of the
fasteners. Track vibration frequency will increase as the stiffness of fasteners
increase. Therefore, in order to achieve a better vibration reduction effect, a
tuned mass damper system should match the stiffness of the fasteners.
Table 4:
common track
HTMD
Fastener force (kN ).
Fastener stiffness（kN / mm）
6
25
50
42.889
60.503
81.476
21.521
32.62
50.018
The first-order frequency of the track with HTMD is about 20Hz, which is
close to that of a light floating slab track (FST). This structure (HTMD) makes
full use of existing railway structure space, which can be used in line
transformation.
4 Wheel-set drop simulation
As a train runs through the track with irregularities on the rail surface, the wheel
tread may separate with the top of the rail. Because the springs under the bogie
WIT Transactions on The Built Environment, Vol 128, © 2012 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
690 Urban Transport XVIII
are in a compression state, when the wheel does not contact with the rail, the
compressed spring will push the wheel bounce to the rail, which causes the
wheel/rail impact. Based on the principle, we use the wheel-set drop method to
simulate. Wheel-set fall down freely from a certain height will make wheel-set
impact vertically to rail, which makes the track structure vibrate. Through the
attenuation of rail impact amplitude and all the parts of the track vibration, we
determine the effect of track structure stiffness and damping parameters on
vibration attenuation characteristics [6].
The parameters of rail and track are in Table 1. The wheel parameters are as
follows:
The wheel model: PLANE42 elements, grid partition using mapping way,
quadrilateral grid shape. Take wheel thickness of 0.0627 m , 1.25 m in diameter.
The wheel-set fall height is taken at 20 mm . This height can simulate the
wheels’ bounces on the rail irregularities state.
Figure 3:
Figure 4:
Displacement of rail (common track).
Displacement of rail (HTMD, k=6 kN / mm ).
WIT Transactions on The Built Environment, Vol 128, © 2012 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
Urban Transport XVIII
Figure 5:
Displacement of rail (HTMD, k=25 kN / mm).
Figure 6:
Displacement of Rail (HTMD, k=50 kN / mm ).
691
Fastener stiffness has a great influence on the vibration reduction
performance. The lower the stiffness of the fasteners, the bigger vertical
displacement the rail is. At the same time the smaller the fasteners’ force is.
Tuned mass has a certain effect on the whole system. As the increase of tuned
mass in TMD, the vibration reduction effect is better. Comparing HTMD with
WIT Transactions on The Built Environment, Vol 128, © 2012 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
692 Urban Transport XVIII
common track and TMD systems, both the fasteners force and displacement of
rail have a greater degree of vibration attenuation.
5 Wheel/rail response result analysis under dynamic load
From Figures 7–10, acceleration curves show that the peak acceleration of rail
and track bed of TMD is reduced when compared with common track. Track
with HTMD has greater reduced amplitude than the other two tracks.
Figure 7:
Rail acceleration history (common track and track with TMD）
.
Figure 8:
.
Rail acceleration history (HTMD）
WIT Transactions on The Built Environment, Vol 128, © 2012 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
Urban Transport XVIII
693
Figure 9:
Track bed acceleration histories (common track and track with
TMD).
Figure 10:
Track bed acceleration history (track with TMD and track with
HTMD).
The analysis results indicate that when the stiffness of fasteners decreases, the
track vibration level decreases. Z vibration level of track bed in common track is
66–69 dB; Z vibration level of track bed in HTMD system is 55–60 dB, which
shows that the HTMD system has a remarkable effect.
WIT Transactions on The Built Environment, Vol 128, © 2012 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
694 Urban Transport XVIII
Table 5:
common track
HTMD
Vertical vibration level.
Fastener stiffness
( kN / mm )
Vertical
vibration
level（dB）
6
25
50
6
25
50
111.70
106.38
102.61
91.01
87.74
84.65
6 Conclusion
Through modal analysis to the common track and heavy mass tuned system track
structure, the relationship between stiffness of fasteners and natural frequencies
has been obtained. When the common fasteners are used, the first order
frequency of HTMD system is 20 to 30 Hz. The first 6 orders are within 40 Hz.
When adopting high elastic fasteners, the first order of HTMD track is below
20Hz, which is close to light FST. The subway can have a large degree of
vibration reduction, which is a highly efficient method to both operating lines in
the reconstruction.
The wheel-set drop simulation shows the parameters have an influence on
track vibration reduction. Analysis results show that lower fastener stiffness have
good contribution to vibration control. Under the same impact height, the rail
vertical displacement reduces by nearly half compared with common track, so as
fasteners force.
The dynamic load test shows that the vibration level of track bed has 5 dB
reduced when using different stiffnesses of fasteners. Compared with common
track, the vibration level of HTMD track reduces by about 15dB. It can be
regarded as a good vibration reduction rail structure.
Acknowledgements
This work was financially supported by the Fundamental Research Funds for the
Central Universities (Tongji University), National Science and Technology
Support Plan (2009BAG11B02).
References
[1] Chen, Y., & Huang, Y., Timoshenko beam with tuned mass dampers and its
design curve. Journal of Sound and Vibration, 278: pp. 873-888, 2004.
WIT Transactions on The Built Environment, Vol 128, © 2012 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)
Urban Transport XVIII
695
[2] Wang, J. F., Lin, C. C. & Chen, B. L., Response to comments on:
“Vibration Suppression for High Speed Railway Bridges Using Tuned
Mass Dampers”. International Journal of Solids and Structures, 42:
pp. 6520-6521, 2005.
[3] Kanto, Y., & Y.G. A dynamic contact buckling analysis by the penalty
finite element method. International Journal Numerical Methods
Engineering, 29 (3), pp. 755-774, 1990.
[4] Ludvigh, E, Elastic Behaviors of Continuously Embedded Rail systems.
Periodical Polytechnica, 46 (1), pp. 103-114, 2002.
[5] Harris, C.M., & Piersol, A.G., Harris’s Shock and vibration handbook,
McGraw- Hill Companies, Inc., pp. 159-167,2008.
[6] Wang Q., Research on track impact using wheel-set drop method. Journal
of the China Railway Society, 14(3), pp. 102-109, 1992.
WIT Transactions on The Built Environment, Vol 128, © 2012 WIT Press
www.witpress.com, ISSN 1743-3509 (on-line)